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December  2015, 35(12): 5769-5786. doi: 10.3934/dcds.2015.35.5769

On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations

Matteo Cozzi 1,

1. 

Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy

Received  March 2014 Published  May 2015

In this brief note we study how the fractional mean curvature of order $s \in (0, 1)$ varies with respect to $C^{1, \alpha}$ diffeomorphisms. We prove that, if $\alpha > s$, then the variation under a $C^{1, \alpha}$ diffeomorphism $\Psi$ of the $s$-mean curvature of a set $E$ is controlled by the $C^{0, \alpha}$ norm of the Jacobian of $\Psi$. When $\alpha = 1$ we discuss the stability of these estimates as $s \rightarrow 1^-$ and comment on the consistency of our result with the classical framework.
Keywords: $C^{1, Fractional mean curvature, \alpha}$ diffeomorphisms, fractional Laplacian, quantitative Implicit Function Theorem., non-local equations, $s$-perimeter.
Mathematics Subject Classification: Primary: 35R11, 28A75, 49Q05; Secondary: 26B1.
Citation: Matteo Cozzi. On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5769-5786. doi: 10.3934/dcds.2015.35.5769
References:
[1]

L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403. doi: 10.1007/s00229-010-0399-4.  Google Scholar

[2]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815. doi: 10.1080/01630563.2014.901837.  Google Scholar

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B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639. Google Scholar

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X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

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X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar

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L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.  Google Scholar

[12]

L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4.  Google Scholar

[13]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240. doi: 10.1007/s00526-010-0359-6.  Google Scholar

[14]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871. doi: 10.1016/j.aim.2013.08.007.  Google Scholar

[15]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[16]

S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790. doi: 10.3934/dcds.2013.33.2777.  Google Scholar

[17]

A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., published online, (2015). doi: 10.1515/crelle-2015-0006.  Google Scholar

[18]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.  Google Scholar

[19]

M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.  Google Scholar

[20]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.  Google Scholar

[21]

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.  Google Scholar

[22]

G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718. doi: 10.1007/s10231-011-0243-9.  Google Scholar

[23]

O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500. doi: 10.1016/j.anihpc.2012.01.006.  Google Scholar

[24]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39. doi: 10.1007/s00526-012-0539-7.  Google Scholar

[25]

E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23. doi: 10.1007/s00032-013-0199-x.  Google Scholar

show all references

References:
[1]

L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403. doi: 10.1007/s00229-010-0399-4.  Google Scholar

[2]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815. doi: 10.1080/01630563.2014.901837.  Google Scholar

[3]

G. Albanese, A. Fiscella and E. Valdinoci, Gevrey regularity for integro-differential operators, J. Math. Anal. Appl., 428 (2015), 1225-1238. doi: 10.1016/j.jmaa.2015.04.002.  Google Scholar

[4]

G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.  Google Scholar

[5]

G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284. doi: 10.1017/S0956792598003453.  Google Scholar

[6]

B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639. Google Scholar

[7]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[8]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar

[9]

L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331.  Google Scholar

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar

[11]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.  Google Scholar

[12]

L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4.  Google Scholar

[13]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240. doi: 10.1007/s00526-010-0359-6.  Google Scholar

[14]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871. doi: 10.1016/j.aim.2013.08.007.  Google Scholar

[15]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[16]

S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790. doi: 10.3934/dcds.2013.33.2777.  Google Scholar

[17]

A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., published online, (2015). doi: 10.1515/crelle-2015-0006.  Google Scholar

[18]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.  Google Scholar

[19]

M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.  Google Scholar

[20]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.  Google Scholar

[21]

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.  Google Scholar

[22]

G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718. doi: 10.1007/s10231-011-0243-9.  Google Scholar

[23]

O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500. doi: 10.1016/j.anihpc.2012.01.006.  Google Scholar

[24]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39. doi: 10.1007/s00526-012-0539-7.  Google Scholar

[25]

E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23. doi: 10.1007/s00032-013-0199-x.  Google Scholar

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